General Cauchy–Lipschitz theory for Δ-Cauchy problems with Carathéodory dynamics on time scales

Abstract

The aim of this paper was to complete some aspects of the classical Cauchy–Lipschitz (or Picard–Lindelöf) theory for general nonlinear systems posed on time scales. Despite a rich literature on Cauchy–Lipschitz type results on time scales, most of the existing results are concerned with rd-continuous dynamics (and -solutions) and do not cover the framework of general Carathéodory dynamics encountered for instance in control theory with measurable controls (which are not rd-continuous in general). In this paper, our main objective was to study -Cauchy problems with general Carathéodory dynamics. We introduce the notion of absolutely continuous solution (weaker regularity than ) and then the notion of maximal solution. We state and prove a Cauchy–Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given -Cauchy problem under suitable assumptions such as regressivity and local Lipschitz continuity. Three new related issues are also discussed in this paper: the boundary value is not necessarily an initial or a final condition, the solutions are constrained to take their values in a non-empty open subset and the behaviour of maximal solutions at terminal points is studied.

Keywords::

• time scale
• Cauchy–Lipschitz (Picard–Lindelöf) theory
• existence
• uniqueness

Keywords::

• 34N99
• 34G20
• 39A13
• 39A12

Notes

^1. Email: [email protected]

^2. Actually, this paper was motivated by the needs of completing the existing results on Cauchy–Lipschitz theory on time scales, in order to investigate nonlinear control systems with measurable controls, and finally to derive a strong version of the Pontryagin maximum principle in optimal control theory on time scales (see [8]).

^3. Indeed, in the discrete case and in the case of an initial condition, such an assertion would imply that an implicit discrete equation is equivalent to an explicit discrete equation. But this is wrong: an implicit equation does not necessarily admit a solution while an explicit equation always does. For example, let us consider and . In this case, the non-shifted -Cauchy problem , , has a unique global solution for any function f. At the opposite, the shifted Δ-Cauchy problem , , has no solution whenever for example. Hence, this shifted problem cannot be reduced into an equivalent non-shifted problem. It can be noted that the reduction procedure mentioned in [19] is based in a crucial way on a regressivity assumption (denoted by (A1^σ) in this paper) on f. We insist that we do not make such an assumption in our paper.